The distance of the point $\mathrm{P}(3,4,4)$ from the point of intersection of the line joining the points $\mathrm{Q}(3,-4,-5)$ and $\mathrm{R}(2,-3,1)$ and the plane $2 \mathrm{x}+\mathrm{y}+\mathrm{z}=7$, is equal to ____

JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Three Dimensional Geometry

The distance of the point

P (3, 4, 4)

from the point of intersection of the line joining the points

Q (3, - 4, - 5)

and

R (2, - 3, 1)

and the plane

2 x + y + z = 7

, is equal to ____

Answer: 7

Solution

The line through

Q (3, - 4, - 5)

and

R (2, - 3, 1)

has direction vector

QR = (- 1, 1, 6)

. Parametric form:

(3 - t, - 4 + t, - 5 + 6 t)

. Substitute into plane equation:

2 (3 - t) + (- 4 + t) + (- 5 + 6 t) = 7 \Rightarrow 6 - 2 t - 4 + t - 5 + 6 t = 7 \Rightarrow 5 t - 3 = 7 \Rightarrow t = 2

. Intersection point:

(3 - 2, - 4 + 2, - 5 + 12) = (1, - 2, 7)

. Distance from

P (3, 4, 4)

(1, - 2, 7)

(3 - 1)^{2} + (4 - (- 2))^{2} + (4 - 7)^{2} = 4 + 36 + 9 = 49 = 7

JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Integer answer

Practise more: all JEE Main PYQs on Three Dimensional Geometry →

More JEE Main questions on Three Dimensional Geometry

Let $(α, β, γ)$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5, 4, 2)$ on the line $r = (- \hat{i} + 3 \hat{j} + \hat{k}) + λ (2 \hat{i} + 3 \hat{j} - \hat{k})$ .
Then the length of the projection of the vector …
JEE Main 2026View solution →
Let the line L pass through the point $(- 3, 5, 2)$ and make equal angles with the positive coordinate axes. If the distance of L from the point $(- 2, r, 1)$ is $\frac{14}{3}$ , then the sum of all possible values of $r$ is :
JEE Main 2026View solution →
Let the line $L_{1}$ be parallel to the vector $- 3 \hat{i} + 2 \hat{j} + 4 \hat{k}$ and pass through the point $(2, 6, 7)$ and the line $L_{2}$ be parallel to the vector $2 \hat{i} + \hat{j} + 3 \hat{k}$ and pass through the point $(4, 3, 5)$ . If the line $L_{3}$ is parallel to the vector …
JEE Main 2026View solution →
Let $P (α, β, γ)$ be the point on the line
$\frac{x - 1}{2} = \frac{y + 1}{- 3} = z$ at a distance $414$ from the point $(1, - 1, 0)$ and nearer to the origin. Then the shortest distance, between the lines …
JEE Main 2026View solution →
If the image of the point $P (1, 2, a)$ in the line $\frac{x - 6}{3} = \frac{y - 7}{2} = \frac{7 - z}{2}$ is $Q (5, b, c)$ , then $a^{2} + b^{2} + c^{2}$ is equal to
JEE Main 2026View solution →
Let L be the line $\frac{x + 1}{2} = \frac{y + 1}{3} = \frac{z + 3}{6}$ and let S be the set of all points $(a, b, c)$ on L, whose distance from the line $\frac{x + 1}{2} = \frac{y + 1}{3} = \frac{z - 9}{0}$ along the line L is 7. Then $\sum_{(a, b, c) \in S} (a + b + c)$ is equal to :
JEE Main 2026View solution →